Collective Probabilistic JudgementsAuthor(s): Salvador Barberá and Federico ValencianoSource: Econometrica, Vol. 51, No. 4 (Jul., 1983), pp. 1033-1046Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/1912050 .

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Econometrica, Vol. 51, No. 4 (July, 1983)

COLLECTIVE PROBABILISTIC JUDGEMENTS

BY SALVADOR BARBERA AND FEDERICO VALENCIANO'

This paper explores the possibilities opened up by combining preference aggregation and randomization in passing from individual to collective judgements about alternatives. We study the distribution of power under functions which assign to each profile of individual preferences a probabilistic judgement on each pair of alternatives x, y, that is, a probability of x being considered socially at least as good as y. Conditions are described under which the power of coalitions to guarantee that an alternative is not declared worse than another is subadditive, while their power to guarantee that one alternative is declared better than another is superadditive. These results extend some of the main findings on the structure of decisive coalitions under deterministic social choice rules.

1. INTRODUCTION

A LONG LIST OF WORKS in the tradition of Arrow's "General Possibility Theo- rem" [1] have established that one cannot aggregate sets of preorders into binary relations without violating some apparently reasonable conditions, as soon as minimal consistency requirements are imposed upon the aggregate relations. When these relations are required to always be transitive (Arrow's social welfare functions), then all power must be concentrated on one individual; if they are to be quasitransitive, the power to impose preferences is in the hands of an oligarchy, each of whose individual members possesses veto power; if they are to be acyclic, some individuals will still usually have veto power (although some additional requirements are needed to qualify this statement). Even in the limit case where society is modeled as if composed of an infinite number of individu- als, power must still be rigidly distributed (around an "invisible dictator" in the case of social welfare functions).

One may want to explore the possibility of giving groups and individuals a better chance of influencing collective decisionmaking by taking into account individual preferences differently. One such way consists of allowing for nonde- terministic collective judgements. Rather than looking for unqualified statements of social preference or indifference between alternatives we may find statements such as the following to reflect acceptable forms of compromise: "given the preferences of individuals, x should be given a 50-50 chance of being declared as good as y;" or, "x should be chosen over y nine out of every ten times."

A number of recent articles have extended the traditional framework of social choice theory, by considering functions whose images are lotteries over binary relations, rather than deterministic "social" preferences. Barbera and Son- nenschein [3], and McLennan [9] studied the case where the social image of individual preferences is given by lotteries over transitive binary relations.

'The authors are grateful to Professors Keiding, Bandyopadhyay, Deb, and Pattanaik for making their papers available before publication. The comments of our colleagues F. and J. Grafe were of great help.

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1034 S. BARBERA AND F. VALENCIANO

Bandyopadhyay, Deb, and Pattanaik [2] extended the results to cover the case of lotteries over quasitransitive relations. Keiding [71 first remarked the generality of the results for infinite societies.

This paper explores the possibilities opened up by combining aggregation and randomization within a broader framework, that of collective probabilistic judge- ment functions. We define a probabilistic judgement on a set of alternatives as a function which assigns a number between zero and one to any ordered pair of alternatives (x, y). This number is to be interpreted as the probability with which x will be declared at least as good as y, according to the probabilistic judgement under consideration. Individual preferences are taken to be preorders on the set of alternatives, and a preference profile is a function assigning one such preorder to each individual in society. Collective probabilistic judgement functions, then, assign a probabilistic judgement to each preference profile.

Our results are about the power of coalitions under collective probabilistic judgement functions satisfying natural extensions of the traditional binarity and Paretian conditions, plus a number of consistency conditions over the probabili- ties which can be assigned to different pairs of alternatives. Two types of power of coalitions are defined, and both are proven to be independent of the pair of alternatives considered and monotonically increasing with the coalition's size. Conditions are described under which a coalition's power to guarantee that an alternative is not declared worse than another is subadditive; while its power to guarantee that one alternative is declared better than another is superadditive.

Our consistency requirements are weaker than those considered in the preced- ing literature, and they enable us to stress the significance of the different restrictions employed. In particular, all of our conditions are necessary but by no means sufficient to guarantee that the probabilistic judgements under consider- ation could be rationalized as resulting from lotteries over quasitransitive prefer- ence relations. Our results generalize those in [21 in this respect, as well as in others. We also cover the case of infinite voters, and we find that additional restrictions arise when the number of alternatives is six or more-an interesting differential fact in the probabilistic framework, with no parallel in the determinis- tic case. This suggests the possibility for further work regarding nonrationalizable procedures (paralleling the deterministic literature on nonrationalizable choice functions), and procedures whose aggregate judgements exhibit different types of rationality (like, for example, those considered in mathematical psychology).

The results in the paper also extend a number of theorems in the deterministic social choice literature. In particular, we see how the important fact that decisive sets constitute filters in the deterministic setup follows from the results presented here, as a consequence of the subadditive nature of our power functions.

Some final words about motivation. It is clear that, unlike in the deterministic framework, there is no obvious connection between the probabilistic judgements of a society and its choices when faced with feasible sets containing more than two alternatives. Those readers who feel uneasy about this general lack of correspondence between judgements and choice may note that the conditions

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COLLECTIVE JUDGEMENTS 1035

laid out in our paper are necessary for different types of rationalizability, under which a clear connection is established between the two. Then, our presentation can be viewed as a set of results on collective probabilistic choice functions, where emphasis is placed on binary choice probabilities, and where an effort is made to identify and separate as much as possible the consequences of different conditions which, taken together, would result from a rational choice hypothesis. However, we feel there is more to our distinction. In a deterministic framework, the point has been strongly made that, although one may establish a connection between judgements and choices, there are cases where the two should be clearly distinguished (for example, in Sen [11]). What we supply is a framework where this distinction could be made in a world admitting both stochastic judgements and choices. The lack of a trivial connection between one and the other appears then as an intriguing problem rather than a disqualifying objection.

2. PROBABILISTIC JUDGEMENTS

Let A be a set, to be called the set of alternatives. A probabilistic judgement on A is a function r: A x A -* [0, 1], such that (a)

(Vx E A) r(x,x) = 1, and (b) (Vx, y ( A) r(x, y) + r(y,x) > 1. A probabilistic judgement thus assigns a number in the real interval [0, 1] to

each ordered pair of alternatives (x, y). This number can be interpreted as the probability with which, according to this judgement, x should be considered at least as good as y. We remark that the standard way to express judgements by means of binary relations can be viewed as a special type of probabilistic judgement, where the only admissible values for r(., *) are zero and one. For this particular subclass, conditions (a) and (b) require that the relations be reflexive and complete, respectively.

Let 5 be the set of all possible judgements on A. One may want to impose some "consistency" requirements on a probabilistic

judgement by demanding that certain relationships hold among the probabilities assigned to different pairs of alternatives. We now proceed to present some such possible requirements.

Let - be the set of binary relations on A which are (i) reflexive, i.e., (Vx E A) (VB E X) xBx; (ii) complete, i.e., (Vx, y E A) (VB E X) [xBy VyBx]. Given B E X, its asymmetric component B is defined so that, for all x, y E A, xBy X [xBy A -yBx].

Let R be the set of binary relations on A which are complete, reflexive, and (iii) transitive, i.e., (Vx, y, z E A) (VB E !) [xBy A yBz] -* xBz.

Let 0 be the set of binary relations on A which are complete, reflexive, and (iv) quasitransitive, i.e., such that their asymmetric component B is transitive. In terms of B, this property requires that (Vx, y, z E A) (VB E )) xBy -* xBz V zBy.

A probabilistic judgement r is stochastically rationalizable iff there exists a probability space [X, B, Pr] such that, for all x, y E A, the set {B E - I xBy} is

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1036 S. BARBERA AND F. VALENCIANO

measurable and r(x, y) = Prr[tB E 9 I xBy}]. If, in addition, v421 = 1, or Pr = 1, we say that r is quasi-transitive or transitive stochastically rationalizable respectively.

The conditions of transitive or quasitransitive stochastic rationalization impose effective restrictions upon the possible values of a probabilistic judgement. A complete characterization of such restrictions has not, to our knowledge, been achieved. However, the following restrictions on r are necessary for quasitransi- tive (and thus, for transitive) stochastic rationalizability. We state them here and prove their necessity in the Appendix.

CONDITION 1 (Consistency under Complete Rejection):

(Vx, y,z E A) r(y,z) = 0->[r(x,z) < r(x, y) A r(y,x) < r(z,x)].

This condition requires that, if y is never judged as good as z, then x should be judged at least as good as y with a greater or equal probability than x at least as good as z.

CONDITION 2 (Stochastic Quasitransitivity over Triples):

(Vx, y,z E A) r(x, y) < r(x,z) + r(z, y).

Our second condition implies the first: for any three alternatives x, y, z, the probability that x be judged at least as good as y should not exceed the sum of the probabilities that x be at least as good as z and that z be at least as good as y.

CONDITION 3 (Stochastic Quasitransitivity over 6-tuples):

(Vx, y,z, u,v, w E A)

[r(x, u) = r(x, w) = r(z, w) = r(z, y) = r(v, u) = r(v, y) = 0]

> [ r(v, w) + 1 < r(y, x) + r(u, z) ].

Unlike its deterministic counterparts, the requirements of stochastic transitive and quasitransitive rationalizability go beyond restrictions involving triples of alternatives. Condition 3 is the simplest of the additional limitations derived from stochastic quasitransitive rationalizability which only apply when the set of alternatives is sufficiently large.

3. COLLECTIVE PROBABILISTIC JUDGEMENT FUNCTIONS

Let I be a set, to be called the set of individuals. A preference profile is a function s: I- -a. It is to be interpreted as a

description of the state of opinion in the society composed of the individuals in I

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COLLECTIVE JUDGEMENTS 1037

regarding the alternatives in A. Each individual i's preferences in s are given by s(i), and s(i) stands for the assymetric component of s(i).

Let / denote the set of all possible preference profiles. A collective probabilistic judgement function is a function of the form f: - 5 ; that is, a function which assigns a probabilistic judgement on A to each

possible profile of preferences on A. More explicitly: for any given profile and any pair of alternatives, a collective probabilistic judgement function determines a number in the real interval [0, 1]. This number is to be interpreted as the probability with which society should declare x to be at least as good as y, given the preferences of its individuals.

def When this does not lead to confusion, we let rS -f(s).

A collective probabilistic judgement functions f is Paretian iff

(Vs E vI) (Vx,y E A) [ (Vi E I) xg(i)y -rs(y,x) = O0.

A collective probabilistic judgement function is binary iff

(Vs, s' E/ ') (Vx, y E A)

[(Vi E I) (xs(i)y = xs'(i)y) - rs(x, y) = rs,(x, y)].

A collective probabilistic judgement function f is transitive (respectively qua- sitransitive) rationalizable iff for all profiles s its image is transitive (respectively quasitransitive) stochastically rationalizable.

4. THE DISTRIBUTION OF POWER UNDER COLLECTIVE PROBABILISTIC JUDGEMENT FUNCTIONS

This section explores the consequences of imposing Conditions 1, 2, and 3 on the images of binary, Paretian, collective probabilistic judgement functions. Although there are no compelling reasons why in general such a function should satisfy these conditions, they are clearly important in the case where we view probabilistic judgements as probabilities of choice over pairs. Then, the condi- tions express a part of the overall restrictions imposed by a rationalizability requirement and the theorems hold, in particular, for quasitransitive rational functions.

Also, even if we do not make any connection between judgements and choices, the results can be seen as a first exploration, one of whose results is to show that many features of those aggregation procedures whose images are stochastically rationalizable will be inherited by others satisfying much milder conditions.

THEOREM 1: Let f be a binary and Paretian collective probabilistic judgement function, whose images always satisfy Condition 1. Then, there exists a function 11: 21-[0, 1], such that for all s E vf, and all x, y E A,

(i) rs(x, y) 2 {i E I|xs(i)y}, and

(ii) rs(x, y) = A{i E IIxs(i)y} whenever (Vi ElI) [xs(i)y V ys(i)x]

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1038 S. BARBERA AND F. VALENCIANO

This function ) also satisfies the following properties:

(iii) (0)= 0,

(iv) i.(I)= 1,

(v) [ Cl 5 C2 5- I] < (CI) < tt(Q-)

The function y can be interpreted as a measure of the power of coalitions. More accurately, it indicates the maximum probability with which a coalition of individuals preferring x over y can guarantee that x will be declared socially as good as y. The main points of the result are that such power is independent of the pair of alternatives under consideration, and that the probability value that a coalition can guarantee itself is exactly attained at those profiles where the complementary coalition's views on a pair of alternatives are strictly opposed.

THEOREM 2: Let f be a binary and Paretian collective probabilistic judgement function whose images satisfy Condition 2. Then, its associated power function is subadditive.

THEOREM 3: Let f be a binary and Paretian collective probabilistic judgement function, and #A > 6. If f's images satisfy Conditions 2 and 3, its associated power function is such that

(dC1,I C2 S- I), [ C1 U C2 = I] --[ II(CI) + I(C2)

- A(Cl n C2) ? 1].

The meaning of the latter result is not transparent, and it may be worth noting why it is important. It has as an immediate corollary (Corollary 3, Section 5) that the strict power of coalitions-that is, their power to impose strict preferences- must be distributed superadditively when there are six or more alternatives. A further consequence of the result is that, when the collective judgements are to be transitive stochastically rationalizable, then the power of coalitions must be distributed additively (Theorem 5.7, Section 7).

The proofs of these theorems, especially 1 and 2, closely parallel Sen's proof of Arrow's theorem, and we just sketch them.

PROOF OF THEOREM 1: Let f be a binary, Paretian collective probabilistic judgement function, whose images always satisfy Condition 1. All statements and definitions refer to this given f.

We say that coalition J C I is k-decisive for x, y E A and write DJ(x, y; k) iff

(Vs Ef) [Jc {i E I I xs(i)y}] - r,(x,y)?k.

We say that coalition J C I is k-quasidecisive for x, y and write DJ(x, y; k)

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COLLECTIVE JUDGEMENTS 1039

iff

(VS E /

[J= {iEII x(i)y} AI-J= {iEIIy9(i)x}]--rs(x,y)?k.

We define the decision power of coalition J for x, y E A as

A*(J; x,y) _ supk Dj(x, y; k)}

It is clear from the definitions that

(i) Di(x, y; k) ->Dj(x, y;k),

(ii) {DJ(X,y;k)Ak'E [0,k]}-Dj(x,y;k'), and

(iii) {Dj(x,y;k) A J C J' C I} I-D j(x,y;k).

It will suffice to prove that, for any x, y, u, v E A (x # y, u # v), and all J C I, ,*(J; x, y) = ,u*(J; u, v). This will guarantee the possibility of defining y

independently of the alternatives under consideration. The remaining properties of y will then follow easily.

Considering a profile s where

(a) Jc{iI x(i)z};

(b) (Vj E- J ) [ xs (j)y A y ( j) z]

(c) (Vdi E- I - [ygs(i)x A ys(i)z ],

we can show that, if a coalition J is k-quasidecisive for x, y E A, it is k-decisive for x, z, where z is any alternative in A distinct from x and y. From this basic observation and a standard social choice-theoretic argument, we arrive at the theorem's conclusions.

PROOF OF THEOREM 2: Since Condition 2 implies Condition 1, we can assert the existence of a function : 2' -- [0, 1] with the properties stated by Theorem 1. We want to show that, in addition,

(VC1 I C2 C I) A(C1 U C2) < A(C1) + A(C2)-

To prove it, suppose not, i.e., 3C1, C2 c I for which A(C1 U C2) > L(C1) + p(C2). Since y(C1 fn C2) < (C1), it will suffice to consider the case where C nl C2 = 0. Take the profiles where

(Vi E C1) xs(i)ys(i)z,

(Vh E I [ C2 ])xs j( )y(

(V/h E- I -[ Cl U C21]) ys (h)zs (h)x.

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1040 S. BARBERA AND F. VALENCIANO

Given the properties of A, r,(x, y) = AL(C1 U C2) > Ai(C1) + AL(C2) = r,(x,z) + r,(z, y), contradicting the assumption that all images of f satisfy Condition 2.

PROOF OF THEOREM 3: We can assert the existence of a function A: 21' [O, 1] with the properties stated by Theorems 1 and 2. We want to show that, when the number of alternatives is at least six, /(C1) + /(C2) 2 1 + /(Cl n C2) for all C1, C2 such that C1 U C2 = I. We need only prove it for the case where C1 U C2 #1 0. Suppose not, i.e., 3C1, C2 C I such that C1 n C2 # 0 and AL(C1) + /(C2) < 1 + /(Cl n C2). Consider the profile s where

(Vi E I - C1) w9(i)u9(i)x9(i)y9(i)v9(i)z,

(VJ E I - C2) ys(jws(j)zs(j)u9(j)x9(j)V,

(Vh E C1 n C2) us(h)ys(h)vs(h)ws(h)zs(h)x.

Given the properties of A,

rs(y,x) + rs(u,z)

=- [(I- C2) U (Cl n C2)] + [(I- C1) u (C in C2)]

= (C) + A(C2) < 1 + (Cl n C2) = 1 + rs(v,w), while

rS (x, u) = rS (x, w) = rS (z, w) = rS (z, y) = rS (v, u) = rS (v, y) = 0,

since f is Paretian. This contradicts the assumption that all images of f satisfy Condition 3.

5. STRICT PROBABILISTIC JUDGEMENTS

A strict probabilistic judgement on A is a function p : A x A -* [0, 1], such that (a) (Vx E A) P(x,x) = 0, and (b) (Vx, y E A) P(x, y) + P(y,x) < 1. A strict probabilistic judgement assigns a number in the real interval [0, 1] to each ordered pair of alternatives (x, y). This number will be interpreted as the probability with which x should be considered strictly better than y according to this strict judgement.

We could now proceed to discuss "consistency" requirements on strict proba- bilistic judgements, and define strict probabilistic judgement functions as an alternative way to process sets of individual preferences. Rather than doing that, we exploit the existence of a natural bijection between strict and non-strict probabilistic judgements, which parallels the usual relationship between a com- plete, reflexive binary relation and its assymetric component.

Given a probabilistic judgement r on A, its associated strict probabilistic judgement T is defined so that

def

(Vdx, y E A) Tr(x, y) =_ I1- r(y, x).

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COLLECTIVE JUDGEMENTS 1041

Given a collective probabilistic judgement function f, its associated strict counterpart f' is defined so that (Vs E /) f'(s) = Ts, where Ts is the strict probabilistic judgement associated to rs = f(s). These relationships are instru- mental: they would allow us to discuss the properties of functions which take preference profiles into strict probabilistic judgements, once we know those of collective probabilistic judgement functions, even if we wanted to consider these two objects as completely independent from each other conceptually. Our treatment of these new functions as derived objects can be justified by the following.

REMARK: For any collective probabilistic judgement function f and any pro- file s, if f(s) = rS is stochastically rationalized by Pr, then f'(s) = TS is also sto- chastically rationalized by the same Pr, in the sense that (Vx, y) r(x, y) =

Pr { B Ec I xBy}. We now can state as corollaries to our previous theorems what we know about

the power of coalitions to strictly impose their strict preferences under collective probabilistic judgement functions.

COROLLARY 1: Let f be a binary and Paretian collective probabilistic judgement function, whose images always satisfy Condition 1. There exists a function q : 2' -* [0, 1], to be called the strict power function for f, such that for all s E Y and all x,y &A,

(i) TS (x, y) 2 ? { i E Ixs (i)y}, and

(ii) Ts(x,y) = q{i E Ilxs(i)y} whenever (V i ElI) [xs(i)y Vys(i)x]

In addition,

(iii) 0() ;

(iv) n I)=1;

(v) [C CC C2 C I] [q(CI) < 'q(C2)1.

COROLLARY 2: Let f be a binary and Paretian collective probabilistic judgement function, whose images always satisfy Condition 2. Then, its associated strict power function 7 is such that

(VCI I C2 E I) 7(CI) + 71 (C2) -7(C1 n C2) < 1

COROLLARY 3: Let # A > 6, and f be a binary and Paretian collective probabi- listic judgement function, whose images always satisfy Conditions 2 and 3. Then, its associated strict power function is superadditive.

A summarizing statement for quasitransitive rational functions is left to the reader.

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1042 S. BARBERA AND F. VALENCIANO

PROOF OF COROLLARY 1: Sincef satisfies the conditions of Theorem 1, we can define q so that (VC c I), 7q(C) = 1 - pt(I - C), where A is the power function associated to f.

To check that (ii) holds, let x, y be any two alternatives and s any profile such that (Vi E I) [xs(i)y V ys(i)x]. From the properties of I,

Ts (x, y) = 1- rs (y, x) = 1- { i I ys(i)x} = 'q { i I xs(i)y}.

To prove that (i) holds, suppose not. Then, there would be alternatives x, y and a profile s such that Ts(x, y) < q { i I xs(i)y}. Since f is binary, we can assume without loss of generality that for some z E A and all i E I, xs(i)y -* xs(i)z A zs(i)y, while ys(i)x -* zs(i)y (and thus ys(i)x -* zs(i)x). For this profile,

r(y,x) = 1 - (x, y) > 1 - {i Ixs(i)y} = .( (i I ys(i)x}

= -u { i I zs(i)x} = r(z,x).

This is a contradiction, since f is Paretian (and thus r(y, z) = 0), and f's images satisfy Condition 1, requiring that r(y, x) < r(z, x). Properties (iii), (iv), and (v) are immediate consequences of the relationship between y and q. Their proofs, as well as those of Corollaries 2 and 3, are left to the reader.

6. THE NUMBER OF INDIVIDUALS; FINITE SOCIETIES

The results obtained up to here are independent of whether society is com- posed of a finite or infinite number of individuals. We can further sharpen them for the case where I is finite.

THEOREM 4: Let f be a binary Paretian collective probabilistic judgement func- tion, whose images always satisfy Condition 2. If I is finite, there exists a unique coalition J C I for which (i) A(J) = 71(J) = 1 and (ii) (VC C J, C #Z 0) pi(C) > 0 hold. Moreover, this unique coalition is such that (iii) (VC C I) A(C) = A(C n J).

PROOF: LetJ = {i E I I j(i) > 0). (i) Suppose 71(J) < 1. Since 71(J) + It(I - J) = 1, we would have p.(I - J) > 0. Since y is subadditive, there would exist some i 4 J such that t(<i>) > 0, a contradiction. Thus, q(J) = 1 and, a fortiori, ML(J)= 1.

(ii) If J' C J and J' M 0, then 3i E J' C J for which 4(<i>) > 0. Thus, (J') >0.

To prove that J as defined is the only coalition for which (i) and (ii) hold, suppose not, and let J' J J be such that q (J') = 1 and (VC C J') I(J) > 0. If J'- J = 0, J' c J, then 1t(J - J') > 0, 4(I - J') > 0, 1(J') = 1 - ,(I - J') < 1, a contradiction to (i). If J'-J - I 0, then 4(I - J) = 1 - 7(J) = 0 and yt(J' - J) = 0, a contradiction to (ii).

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COLLECTIVE JUDGEMENTS 1043

Finally, we prove that (iii) holds. Let C C I. Obviously, A(C) 2 A (J n C). If C - J = 0, the result is trivial. Suppose C - J 0. Then, 0 < A (C - J) < p(I - J) = 1 - 7(J) = 0. Since A is subadditive, (C) < p,(C - J) + pt(C n J)= A(C n J). Thus A(C) = A(C n J).

7. TRANSITIVE STOCHASTIC RATIONALITY

When the images of a collective probabilistic judgement function are required to be transitive stochastically rationalizable, all the preceding results apply. But in this particular case an interesting new phenomenon occurs: the power and the strict power function coincide. We state and prove this fact as a Lemma, and then spell out its implications as Theorem 5.

LEMMA 1: Let f be a transitive rational, binary Paretian collective probabilistic judgement function. Its associated power function A and strict power function 71 are identical.

PROOF: We have already remarked that Au(J) 2 r(J), for all J c I. It will thus suffice to show that, under the present hypothesis, q (J) 2 Au(J) for any J c I. Let J be any such coalition, and consider a profile s E = for which (Vi E J) zs(i)xs(i)y and (Vj e I - J) ys(j)zs(j)x. Since f is Paretian, rs(x,z) = 0 and Ts(z, x) = 1. Since f(s) = rS is transitive stochastically rationalizable, Ts(z, x) < Ts (z, y) + Ts (y, x) (see Appendix). Therefore 1 < Ts (z, y) + Ts (y, x), and A (J) = rs(x, y) < Ts(z, y) = 7 (J). This completes the proof of the Lemma. Remark that, in fact, the condition we have used is much weaker than transitive rationalizability of rs.

The following theorem spells out some consequences of the Lemma upon our preceding results.

THEOREM 5: Let f be a transitive rational binary, Paretian collective probabilistic judgement function. There exists a function q: 2' -* [0, 1] such that

(i) (Vx, y E A) (Vs E d-) rs(x, y) ? Ts(x, y) ? p{ i ECI xs(i)y}.

(ii) (Vlx, y E A) (Vds C ? rs(x, Y) = Ts(x, y) = m { i E I xs(i)y}

whenever (Vi E I) [xs(i)y V ys(i)x].

(iii) m(0) = 0.

(iv) [C1 C C2 C I] -[q)(C1) < p(C2)].

(v) (q. C I) s)(J) + q)(I- J) = 1.

(vi) /lp is subadditive.

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1044 S. BARBERA AND F. VALENCIANO

(vii) If #A > 6, m is additive.

(viii) If I is finite, there exists a unique coaltion J C I such that

p(J) = 1 and q(J') > Ofor all J' C J. This unique

coalition is such that (VJ' C I) p(J') = .p(J' In J).

Point (vii) in the Theorem deserves a comment. The result that the power function under our conditions must be additive when the number of alternatives is six or more was first proved by McLennan in [9]. We feel that our presentation helps to clarify this rather surprising result. Additivity appears here as the result of the fact that, under weaker assumptions, (a) the power function must be subadditive (Theorem 2), (b) the strict power function must be superadditive if #A > 6 (Theorem 3), and the fact that, in the present case (c) both functions must be identical (Lemma 1).

8. SUBADDITIVE POWER AND FILTERS

It is clear that social welfare functions and quasitransitive social decision function can be seen as particular cases of the functions considered here, when the only admissible values for aggregate judgements, are 0 and 1. Arrow's General Possibility Theorem [1] can be obtained from Theorem 5 (see [3]). The presence of oligarchies under quasitransitive social decision functions, a result first proved by Gibbard [6] is obtained from Theorem 4 (see also [2]).

It is known that such privileged groups or individuals as a dictator or an oligarchy need not arise when society consists of an infinite number of voters, but that the decision power will still be tightly concentrated in such cases (around "invisible dictators" or "invisible oligarchies"). Both the finite and the infinite case under deterministic aggregation procedures are essentially identical in that the set of decisive coalitions constitute a filter (an ultrafilter in the transitive case). The properties of the strict power function play a similar though more general role in our case. We end up by proving that indeed, under any quasitransitive rational binary Paretian collective probabilistic judgement function f the set of coalitions whose strict power equals one constitute a filter. If, in addition, the function is transitive rational and the only admissible values for a judgement are 0 or 1, this same set is an ultrafilter. Let q be the strict power function associ- ated to f. Since f is Paretian, 0 X {J C I j 7q(J) = 1 } 3 I. Since f is quasitransi- tive rational, q is increasing. Thus if C E {J C I I 71(J) = 1 } and C' D C, then C' E {J c I j q(J) = 1I}. Since f is quasitransitive rational, (V C, C' c I)rq(C) + q(C')-q(C n C')< 1. Thus, if C,C' C {J c IIq (J) = } then C n C IE {J c I I (J) = I} too. This already proves that {J c IIr (J)= 1) is a filter. If, in addition, f is transitive rational and the only admissible values for the aggre- gate judgements are 0 or 1, then (VC c I)rq(C) + q(I - C) = 1, and C X

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COLLECTIVE JUDGEMENTS 1045

{J C II q (J) = 1} implies that I - C e {J C I I q (J) = l}, thus proving that this set is an ultrafilter.

Universidad del Pais Vasco, Bilbao, Spain

Manuscript received December, 1980; revision received June, 1982.

APPENDIX

LEMMA 2: If a judgement r is quasitransitive stochastically rationalizable, (Vx, y, z E A) r(x, y) < r(x, z) + r(z, y).

PROOF: It is true for any quasitransitive binary relation B that (Vx, y, z E A) xBy -- xBz V zBy. Thus, if vr rationalizes r,

Vr{B E 2 1 xBy} < Vr{B E 1 | xBz} + vr{B E ? 1 zBy},

and r(x, y) < r(x, z) + r(z, y).

LEMMA 3: If #A > 6, r is a quasitransitive stochastically rationalizable judgement and r(x, u) = r(x, w) = r(z, w) = r(z, y) = r(v, u) = r(v, y) = 0, then

r(v, w) + 1 < r(y, x) + r(u, z).

PROOF: It is true for any quasitransitive binary relation that

vBw -> vBy V yBw - vBy V yBx V xBw,

vBw -> vBz V zBw - vBu V uBz V zBw.

Thus, when -vBy, -xBw, ~-vBu, -zBw, it must be that vBw -* (yBx A uBz). Therefore, if vr rationalizes r,

Pr{B E 1 1 vBw} < vP{B E 2 lyBx A uBz}

= Pr{B e 2 jyBx} + Vr{B E 1 1 uBz} - vr{B E 2 lyBx V uBz}.

On the other hand,

uBx - uBz V zBx -> uBz V zBy V yBx.

When -zBy, uBx -* uBz V yBx. Thus,

vr{B E G 1 uBx} < vr{B E J lyBx V uBz}.

Substituting,

Vr(B GE 2 1 vBw) < vP{B E 2 lyBx) + Vr{B GE 21 uBz} -vr{B E 2 1 uBx}.

Since r(u, x) = 1, given that r(x, u) = 0,

r(v, w) < r(y, x) + r(u, z) - r(u, x) = r(y, x) + r(u, z) -.

LEMMA 4: If a judgement r is transitive stochastically rationalizable,

(Vlx, y,z E A) r(x, y) < r(x,z) + r(z, y).

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1046 S. BARBERA AND F. VALENCIANO

PROOF: It is true for any transitive binary relation that (Vx, y, z E A) xBy -> xBz V zBy. Thus, if vr rationalizes r,

vr{B E I XBY} < Vr{B E I xBz} + Vr{B E M I zBy},

and r(x, y) < r(x, z) + r(z, y).

REFERENCES

[1] ARROW, K. J.: Social Choice and Individual Values. New York, Wiley, 1951; 2nd. ed. 1963. [2] BANDYOPADHYAY, T., R. DEB, AND P. PATrANAIK: "The Structure of Coalitional Power Under

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ings," Journal of Economic Theory, 18(1978). [4] FISHBURN, P. C.: "Models of Individual Preference and Choice," Synthese, 36(1977). [5] FISHBURN, P. C.: "Choice Probabilities and Choice Functions," Journal of Mathematical Psychol-

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